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Econometric Analysis of Panel Data

11. Parameter Heterogeneity – “Heterogeneous Panels”

Agenda

 Random Parameter Models

 Fixed effects

 Random effects

 Heterogeneity in Dynamic Panels

 Random Coefficient Vectors-Classical vs. Bayesian

 General RPM Swamy/Hsiao/Hildreth/Houck

 Hierarchical and “Two Step” Models

 ‘True’ Random Parameter Variation

 Discrete – Latent Class

 Continuous

 Classical  Bayesian

Cross Country Growth Convergence

i,t i,t i,t i,t^1 i,t i,t i,t-1 i,t- i,t 1 i i,t 1 i,t i i,t 1

Y K (A L )

A index of technology K capital stock = I + (1- )K I s Y Investment Savings L labor (1 n )L

α −α

− − −

= δ = = = = = +

Solow - Swan Growth Model

Equilibrium Model for Steady i,t i i i i,t 1 i,t i i i i i i

logY t log Y (1- )g , g =technological growth rate "convergence" parameter; 1- = rate of convergence to steady state Cross country comparisons of convergence rat

= μ + θ + λ (^) − + ε θ = λ λ = λ

State Income

es are the focus of study.

Heterogeneous Production Model

Health i,t (^) i iHEXP i,t (^) i EDUCi,t i,t i country, t=year Health = health care outcome, e.g., life expectancy HEXP = health care expenditure EDUC = education Parameter heterogeneity: Discrete? Aids domin

= α + β + γ + ε

ated vs. QOL dominated Continuous? Cross cultural heterogeneity World Health Organization, "The 2000 World Health Report"

Parameter Heterogeneity

it it i i i X i i i i

y

E[ | ] zero or nonzero - to be defined

E [E[ | ]] =

Var[ | ] to be defined, constant or variable

= ′ + ε = +

it i i

Generalize to Random Parameters x β

β β u

u X u X 0 u X

"The Pooling Problem : " What is the consequence

of estimating under the erroneous assumption of constant parameters. (Theil, 1960, "The Aggregation

Problem") (Maddala, 1970s - 1990s, "The Pooling

Problem")

Fixed Effects

(Hildreth, Houck, Hsiao, Swamy)

it it i i i i i i i i i i i X i i X i i i

y , each observation , T observations

Assume (temporarily) T > K. E[ | ] =g( ) (conditional mean) P[ | ] =( -E[ ]) (projection) E [E[ | ]] = E [P[ | ]] = Var[ |

= ′ + ε = + = +

it i i i i i

x β y X β ε β β u

u X X u X X X θ u X u X 0 u X i ] = Γ constant but nonzero

Estimating the Fixed Effects Model

N i 1

Estimator: Equation by equation OLS or (F)GLS (^1) ˆ Estimate? is consistent N =

          (^)        (^) =    (^) +    (^)        (^)        ^     

Σ

1 1 1 1 2 2 2 2

N N N N

i

y X 0 ... 0 β ε y 0 X ... 0 β ε ... ... ... ... ... ... ... y 0 0 ... X β ε

β β for E[ βi ] in N.

Partial Fixed Effects Model

i i

Ni 1 1 Ni 1 (^) i 1 (^1) i

Some individual specific parameters

• , T observations Use OLS and Frisch-Waugh ˆ (^) [ ] [ ], I ( ) ˆ (^) [ ] ( - ˆ) E.g., Individual specific tim

= −^ = − −

= Σ ′^ Σ ′^ = − ′^ ′

= ′^ ′

i i i i

i iD^ i i iD^ i iD i i i i i i i

y D α X β ε

β X M X X M y M D D D D

α D D D y X β

it i0 i1 it

it i0 it

e trends, y t ; Detrend individual data, then OLS E.g., Individual specific constant terms, y ; Individual group mean deviations, then OLS

= α + α + ′ + ε

= α + ′ + ε

it

it

x β

x β

Heterogeneous Dynamic Model

i,t i i i,t 1 i it i,t i i i

logY log Y x

Long run effect of interest is (^1)

Average (over countries) effect: or 1

(1) "Fixed Effects:" Separate regressions, then average results. (2) Country means (o

= μ + λ (^) − + θ + ε β = θ − λ β θ − λ

ver time) - can be manipulated to produce consistent estimators of desired parameters (3) Time series of means across countries (does not work at all) (4) Pooled - no way to obtain consistent est

i i i

imates. (5) Mixed Fixed (Weinhold - Hsiao): Build separate into the equation with "fixed effects," treat u as random.

λ θ = θ +

“Fixed Effects” Approach

i,t i i i,t 1 i it i,t i i i i i i Ni 1 (^) i Ni 1 i i Ni 1 i Ni 1

logY log Y x

1 ,^ = 1

(1) Separate regressions; ˆ, ˆ^ ,ˆ

ˆ 1 ˆ^1 ˆ

(2) Average estimates = ˆ or

N 1 N

ˆ (1 / N) ˆ

Function of averages: =

1 (1 / N)

−

= =

=

β = θ^ β θ

β Σ^ θ

− Σ i

i N i=1 i

In each case, each term i has variance O(1/T )

Each average has variance O(1/N) (1/N)O(1/T )

Expect consistency of estimates of long run effects.

× (^) ∑

Country Means (cont.)

i i i i, 1 i i i i i i T i i i i i i (^) i i (^) T i ii i i i i (^0) i i i i 0 0 i i i i

logY log Y x (logY (y) / T ) x

1 1 x^1 (y) / T 1 (1) Let = /(1 ), expect to be random = /(1 ) and to be random (2) Lev

= μ + λ (^) − + θ + ε = μ + λ − ∆ + θ + ε = μ^ + θ^ − λ^ ∆ + ε − λ − λ − λ − λ δ = μ − λ δ − δ β θ − λ β − β i i T i i i^0

el variable x should be uncorrelated with change (logY (y) / T ).

Regression of logY on 1, x should give consistent estimates of and.

δ β

Time Series

t Ni 1 i,t Ni 1 (^) i i,t 1 Ni 1 (^) i i,t Ni 1 i,t Ni 1 i i i Ni 1 (^) i i,t 1 1,t Ni 1 i i,t 1

log y (1 / N) log y

= + (1/N) y (1/N) x (1/N)

Use =(1/N) so ( ) then

(1/N) log y log y (1/N) ( ) log y

Likewise for (1/N)

= = − = = = = − − = −

Ni 1 i i,t

t (^) 1,t t t Ni 1 i i,t 1

x

log y log y x ( (1/N) ( ) log y ...)

Disturbance is correlated with the regressor. There is

no way out, and by construction no instrumental variable

that could be correlated

= − = −

with the regressor and not the

disturbance.

Mixed/Fixed Approach

N i,t i i 1 i i,t i,t 1 i i,t i,t i,t i i i

log y d log y x

d = country specific dummy variable. Treat and as random, is a 'fixed effect.' This model can be fit consistently by OLS.

= μ + Σ (^) = λ (^) − + θ + ε

μ θ λ

Random Effects and Random Parameters

it it i i i i i i i i

Random Parameters Model y , each observation , T observations

Assume (temporarily) T > K.

E[ | ] =

Var[ | ] constant but nonzero We differentiate the classical and

= ′ + ε = + = +

=

it i i i i i

THE x β y X β ε

β β u

u X 0 u X Γ Bayesian interpretations Randomness here is heterogeneity, not "uncertainty" Bayesian approach to be considered later.